Turbulence in More than Two and Less than Three Dimensions

Antonio Celani

Description

Turbulence in More than Two and Less than Three Dimensions Antonio Celani,1 Stefano Musacchio,2,3 and Dario Vincenzi3 1CNRS URA 2171, Institut Pasteur, 28 rue du docteur Roux, 75724 Paris Cedex 15, France 2Dipartimento di Fisica Generale and INFN, Universita di Torino, via P. Giuria 1, 10125 Torino, Italy 3CNRS UMR 6621, Laboratoire J. A. Dieudonne, Universite de Nice Sophia Antipolis, Parc Valrose, 06108 Nice, France (Received 12 January 2010; published 7 May 2010) We investigate the behavior of turbulent systems in geometries with one compactified dimension. A novel phenomenological scenario dominated by the splitting of the turbulent cascade emerges both from the theoretical analysis of passive scalar turbulence and from direct numerical simulations of Navier- Stokes turbulence. DOI: 10.1103/PhysRevLett.104.184506 PACS numbers: 47.27.Ak, 47.27.ek, 47.27.T In statistical physics most systems display a strong dependence on the space dimensionality. The best known example is the existence of critical dimensions in phase transitions. Among far-from-equilibrium systems, hydro- dynamic turbulence shows a remarkable dependence on the spatial dimension as well. In three dimensions, the nonlinear interaction between different scales is described by the Kolmorogov-Richardson direct cascade: the kinetic energy injected at large scale by an external forcing is transferred to smaller and smaller eddies until it reaches the scales where it is dissipated by viscosity [1].

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• direct cascade


• configuration can


• inverse cascade


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• kinetic energy


• growth rate


• scale dissipation


• dimensional turbulence

Sujets

• Institut Pasteur
• Cambridge University
• Torino
• Princeton University
• Musacchio
• Antipolis

Informations

• Publié par: pefav
• Nombre de lectures: 8
• Langue: English

Extrait

PRL104,184506 (2010)

P H Y S I C A LR E V I E WL E T T E R S

week ending 7 MAY 2010

Turbulence in More than Two and Less than Three Dimensions 1 2,33 Antonio Celani,Stefano Musacchio,and Dario Vincenzi 1 CNRS URA 2171, Institut Pasteur, 28 rue du docteur Roux, 75724 Paris Cedex 15, France 2 Dipartimento di Fisica Generale and INFN, Universita` di Torino, via P. Giuria 1, 10125 Torino, Italy 3 CNRS UMR 6621, Laboratoire J. A. Dieudonne´, Universite´ de Nice Sophia Antipolis, Parc Valrose, 06108 Nice, France (Received 12 January 2010; published 7 May 2010) We investigate the behavior of turbulent systems in geometries with one compactified dimension. A novel phenomenological scenario dominated by the splitting of the turbulent cascade emerges both from the theoretical analysis of passive scalar turbulence and from direct numerical simulations of Navier- Stokes turbulence. DOI:10.1103/PhysRevLett.104.184506PACS numbers: 47.27.Ak, 47.27.ek, 47.27.T

In statistical physics most systems display a strong dependence on the space dimensionality. The best known example is the existence of critical dimensions in phase transitions. Among far-from-equilibrium systems, hydro- dynamic turbulence shows a remarkable dependence on the spatial dimension as well. In three dimensions, the nonlinear interaction between different scales is described by the Kolmorogov-Richardson direct cascade: the kinetic energy injected at large scale by an external forcing is transferred to smaller and smaller eddies until it reaches the scales where it is dissipated by viscosity [1]. By con- trast, in two dimensions, the simultaneous conservation of kinetic energy and enstrophy results in an inverse energy cascade; i.e., the energy injected by the forcing is trans- ferred to large-scale structures [2]. Moreover, three- dimensional turbulence is characterized by anomalous scaling and small-scale intermittency [1], whereas the inverse cascade is apparently self-similar and even shows some signatures of conformal invariance [3]. The transition between the two behaviors and the possible existence of a critical dimension betweend¼3andd¼2have been investigated mainly in models of turbulence where the dimension was introduced as a formal parameter [4–6]. In this Letter, we rather opt for a geometrical way of looking in between integer dimensions, and examine tur- bulent systems which can be regarded as transitional be- tween ad-dimensional isotropic system and a (d1)- dimensional one. Namely, we consider ad-dimensional isotropic system and make one dimension of the space periodic. The compactified dimension can then be col- lapsed or inflated at will so as to connect continuously the two extreme cases. We start by considering the turbu- lent transport of a passive scalar field. This system displays a close similarity to hydrodynamic turbulence (intermittent direct cascade of scalar variance or scale-invariant inverse cascade depending on the properties of the flow), and has the advantage of being analytically solvable [7]. We then study the full hydrodynamical problem by means of direct numerical simulations. The novel feature that emerges from our study is the splitting of the turbulent cascade in both systems. The scalar variance (or the kinetic energy)

0031-9007=10=104(18)=184506(4)

injected at a given length scale flows both toward small and large scales, thus giving rise to a simultaneous double cascade of the same quantity. Let us start with the analysis of the analytically solvable case. The evolution of a passive scalar fieldðx; tÞis described by the advection-diffusion equation: 2 @ t þv r¼rþ’;(1) whereis the diffusivity of the scalar and’ðx; tÞis the source term. The scalar field does not influence the veloc- ity, whose statistical properties are given. We model the turbulent flow by means of the Kraichnan ensemble [8]. Thus,vðx; tÞis a Gaussian stochastic process with zero mean and correlation:hv  ðxþr; tþÞv  ðx; tÞi ¼ 0 0 ½ddÞwit constantanddðrÞ ¼   ðrÞðhd   dðrÞ. The flow is statistically homogeneous, parity in-  variant, stationary, and invariant under time reversal. The tensor2d  ðrÞrepresents the spatial correlation of the velocity differences [7]; its form will be specified later. The source is assumed to be random as well, and more specifically Gaussian, independent of the velocity, with zero mean and correlation:h’ðxþr; tþÞ’ðx; tÞi ¼ ðrÞðÞ, whereðrÞrapidly decays to zero forr¼ krk greater than the correlation length‘ ’ . To study the cascade of scalar variance, we consider the single-time correlationCðr; tÞ  hðxþr; tÞÞðx; tÞi, which does not depend onxowing to the statistical homo- geneity of the velocity. In the Kraichnan model,Cðr; tÞ satisfies the partial differential equation @ t C¼MðrÞCþ;(2) Þ@ @þ2@ @ whereMðrÞ ¼d  ðr r  r  r  r  (summation over repeated indexes is implied) [8]. Without loss of generality, we assumeðx;0Þ ¼0and henceCðr;0Þ ¼0. The (gen- eralized) solution of Eq. (2) then takes the form [9] Z t Z Cðr; tÞ ¼ds dðÞpðr; t;; sÞ:(3) 0 In the above equation,pðr; t;; sÞis the probability density function that two fluid particles being at separationrat

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American Physical Society2010 The